**Amicable Numbers**

A pair of numbers is amicable if each is the sum of the proper divisors of the other. The smallest pair is 220 and 284. The proper divisors of 220 sum to,

1 + 2 + 4 + 5 + 10 + 11 + 20 + 22 + 44 + 55 + 110 = 284

and similarly,

1 + 2 + 4 + 71 + 142 = 220.

According to the philosopher Iamblichus (c. AD 250â€“330), the followers of Pythagoras â€œcall certain numbers amicable numbers, adopting virtues and social qualities to numbers, such as 284 and 220; for the parts of each have the power to generate the other,â€ and Pythagoras described a friend as â€œone who is the other I, such as are 220 and 284.â€

The first few amicable pairs are: (220, 284), (1184, 1210), (2620, 2924), (5020, 5564), (6232, 6368)

In the Bible (Genesis 32:14), Jacob gives 220 goats (200 female and 20 male) to Esau on their reunion. There are other biblical references at Ezra 8:20 and 1 Chronicles 15:6, while 284 occurs in Nehemiah 11:18. These references are all to the tribe of Levi, whose name derives from the wish of Leviâ€™s mother to be amicably related to his father. They were also used in magic and astrology. Ibn Khaldun (1332â€“1406) wrote that â€œthe art of talismans has also made us recognize the marvelous virtues of amicable (or sympathetic) numbers. These numbers are 220 and 284. . . . Persons who occupy themselves with talismans assure that these numbers have a particular influence in establishing union and friendship between individuals.â€ (Ore 1948, 97)

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